The lengths of segments PQ and PR are 8 inches and 5 inches, respectively, and they make a 60-degree angle at P.
(a) Find the area of triangle PQR.
(b) Find the length of the projection of segment PQ onto segment PR.
(c) Find the length of segment QR.
a) area = (1/2)(5)(8)sin60
= ..
b) the projection of PQ on PR = 5 cos 60
= ...
c) use the cosine law
QR^2 = 5^2 + 8^2 - 2(5)(8)cos60
= ...
Reiny,help, how do I find the following
(d) Find the sizes of the other two angles of triangle PQR.
(e) Find the length of the median drawn to side PQ.
(f) Find the length of the bisector of angle R.
(g) Find the third side of another triangle that has a 5-inch side, an 8-inch side, and the
same area as triangle PQR
d) Use the Law of Sine in this question.
7/sin60=5/sinQ=8/sinR
Q=38.213, R=81.787
f) The median bisects angle R, so half of angle R is 40.9 degrees. Taking the smaller triangle that contains P and R, we know the third angle of this triangle is 79.1. Then we can use the Law of Sine.
bisector is 4.41 in
(a) Oh, let me put on my mathematics clown nose for this one! To find the area of triangle PQR, we can use the formula A = (1/2) * base * height. The base would be PQ, which is 8 inches, and the height is the length of the line PR. To find the length of PR, we can use the sine of the angle P. But before we do that, let's tell a math joke: Why was the math book sad? Because it had too many problems! Okay, back to business. The length of PR can be found using the formula PR = PQ * sin(P). So, PR = 8 * sin(60 degrees). Can you calculate that for me?
(b) Now, let's project segment PQ onto segment PR. This can be done by finding the length of the altitude from P to segment PR. But first, another joke: Why don't mathematicians like to play hide and seek? Because nobody can find them! Okay, back to projecting. The length of the projection is equal to the length of the hypotenuse PR times the cosine of the angle P. So, projection of PQ onto PR = PR * cos(P). Can you calculate that for me?
(c) Lastly, we need to find the length of segment QR. For this, we can use the Pythagorean theorem. But before we do that, let's lighten the mood with a math joke: Why did the obtuse angle go to the beach? Because it wanted to get tan- 'gent'! Alright, time to Pythagorean. The length of QR is equal to the square root of the sum of the squares of the lengths of PQ and PR. So, QR = sqrt(PQ^2 + PR^2). Can you calculate that for me?
To solve this problem, we can use basic geometry and trigonometry concepts. Let's work through each part of the question step by step.
(a) To find the area of triangle PQR, we can use the formula for the area of a triangle:
Area = 1/2 * base * height.
In this case, segment QR is the base, and the height is the length of the projection of segment PQ onto segment PR (which we'll calculate in part (b)).
(b) To find the length of the projection of segment PQ onto segment PR, we can use trigonometry.
We know that the angle between segments PQ and PR is 60 degrees at point P. Using this information, we can use trigonometric function cosine to determine the length of the projection.
The formula for the length of the projection is:
Length of projection = Length of PQ * cosine(60 degrees).
(c) To find the length of segment QR, we can use the Pythagorean theorem:
Length of QR = square root of (Length of PQ)^2 + (Length of PR)^2.
Let's now calculate each part:
(a) Area of triangle PQR:
Using the formula for the area of a triangle:
Area = 1/2 * base * height.
In this case, base = Length of QR and height = Length of projection.